Solve 9v^2+81v

$$9 v ^ { 2 } + 81 v$$

Factor

$$9v\left(v+9\right)$$

Solution Steps

Factor out 9.

$$9\left(v^{2}+9v\right)$$

Consider v^{2}+9v. Factor out v.

$$v\left(v+9\right)$$

Rewrite the complete factored expression.

$$9v\left(v+9\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$9v^{2}+81v=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$v=\frac{-81±\sqrt{81^{2}}}{2\times 9}$$

Take the square root of 81^{2}.

$$v=\frac{-81±81}{2\times 9}$$

Multiply 2 times 9.

$$v=\frac{-81±81}{18}$$

Now solve the equation v=\frac{-81±81}{18} when ± is plus. Add -81 to 81.

$$v=\frac{0}{18}$$

Divide 0 by 18.

$$v=0$$

Now solve the equation v=\frac{-81±81}{18} when ± is minus. Subtract 81 from -81.

$$v=-\frac{162}{18}$$

Divide -162 by 18.

$$v=-9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 0 for x_{1} and -9 for x_{2}.

$$9v^{2}+81v=9v\left(v-\left(-9\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$9v^{2}+81v=9v\left(v+9\right)$$

Evaluate

$$9v\left(v+9\right)$$